Universal Life Lapses and Account Depletion

Why Lapse Timing and Account Depletion Matter

A universal life policy has an account value that changes as premiums, loads, interest credits, and charges are applied. That account is not the same as the insurer’s reserve, and a positive account does not by itself determine how many policies remain in force. Death, voluntary lapse, and insufficient account value can each end coverage, but they do so at different points on the projection timeline. A reliable calculation makes those events and their order explicit.

This tutorial uses a discrete annual model. It focuses on how lapse timing and depletion affect exposure to future claims and expenses, and how to reflect those effects in a simple prospective reserve and a scenario-based capital proxy. Any product rules or data in the examples are hypothetical.

Definition: Let $A_{t-1}$ be the account value at the start of policy year $t$, conditional on the policy being in force. Let $P_t$ be the premium received at the start of the year, $\alpha_t$ the premium-load rate, $j_t$ the effective annual account-crediting rate, and $C_t$ the charges deducted at year-end. The calculated account immediately after year-end charges is $$ B_t=\bigl(A_{t-1}+P_t(1-\alpha_t)\bigr)(1+j_t)-C_t. $$ If $B_t\leq 0$, the contract terminates at year-end under the no-grace-period, no-secondary-guarantee convention used here. A contract with a grace period or a no-lapse guarantee requires its own termination rule.

The account recurrence is conditional on the contract being in force at the start of the year. It is not a probability-weighted average across a group: first project an individual contract’s account under the stated path, then apply the contract’s termination rule. If account values or charges differ across policies, depletion may affect only some policies, so the group projection must track that fraction or model the distribution of account values.

Set the Annual Event Order Before Calculating

For the probability calculations below, use this sequence: death occurs during the policy year; among those who survive the year, voluntary lapse occurs at year-end; among survivors who do not voluntarily lapse, account depletion may terminate coverage at year-end. A death benefit is paid for a death during the year. The annual model treats the year’s death probability as applying to policies in force at the start of the year.

Definition: Let $q_t$ be the probability of death during year $t$ for a policy in force at the start of that year. Let $l_t$ be the probability of voluntary lapse at year-end conditional on surviving the year. Let $d_t$ be the probability of account depletion at year-end conditional on surviving and avoiding voluntary lapse. Then the probability of remaining in force at the start of the next year is $$ {}_{{1}}p_t^{\text{in force}}=(1-q_t)(1-l_t)(1-d_t). $$ The product $(1-q_t)(1-l_t)$ is only the probability of surviving and avoiding voluntary lapse. It is the probability of remaining in force only when account depletion does not occur. If depletion is deterministic for every policy in a group, $d_t$ is either 0 or 1; with heterogeneous policies, it can represent the conditional fraction depleted.

The event order is a modeling convention, not a universal property of all contracts. For example, a contract may permit a grace period after insufficient funds, or a lapse may occur during the year rather than at year-end. In those cases, the timing of premiums, charges, claims, and termination must follow the contract terms. In particular, a year-end lapse does not remove a policy from the death exposure it had during that year.

Worked Example: Projecting an Account Before Applying the Depletion Rule

A hypothetical policy starts the year with an account of $\$500$. It receives a $\$100$ premium at the start of the year, with a 10% premium load. The account earns an effective annual crediting rate of 3%. Expense charges of $\$25$ and mortality charges of $\$40$ are deducted at year-end. Calculate the year-end account and determine whether depletion terminates coverage.

Method. Apply the account recurrence in the stated order: deduct the load from the premium, credit interest on the account plus net premium, and then deduct the year-end charges.

The net premium is $$ 100(1-0.10)=\$90. $$ The account before charges is $$ (500+90)(1.03)=590(1.03)=\$607.70. $$ The total year-end charges are $$ 25+40=\$65. $$ Therefore the calculated year-end account is $$ B_t=607.70-65=\$542.70. $$ Since $\$542.70>0$, the account does not deplete under the stated rule, so the policy remains eligible for coverage in the next year, subject to death or voluntary lapse.

Check and common mistake avoided. The load applies to the premium, not to the opening account. The net premium is $\$90$, not $\$100$, and the resulting pre-charge account is $\$607.70$. Deducting $\$65$ gives $\$542.70$, not $\$566.65$.

Why the In-Force Probability Is a Product

The probability formula follows from conditional probability. To be in force at the next year’s start, a policy must survive the year, avoid voluntary lapse given survival, and avoid depletion given both earlier conditions. The probability of a sequence of events equals the probability of the first event multiplied by the conditional probability of the second given the first, and then by the conditional probability of the third given the first two. Thus

\begin{aligned} P(\text{in force next year}) &=P(\text{survive})\\ &\quad\times P(\text{avoid lapse}\mid\text{survive})\\ &\quad\times P(\text{avoid depletion}\mid\text{survive and avoid lapse})\\ &=(1-q_t)(1-l_t)(1-d_t). \end{aligned}

This derivation does not assume that the three termination probabilities are unconditional or independent. The conditional meanings of $l_t$ and $d_t$ are what make the multiplication valid. Using unconditional probabilities in the same product would generally be incorrect.

Worked Example: Separating Death, Lapse, and Depletion

A hypothetical block has 1,000 policies in force at the start of a year. Each has a 2% probability of death during the year. Among those who survive, 8% voluntarily lapse at year-end. Among survivors who avoid voluntary lapse, 5% deplete their accounts at year-end. For each death, the insurer pays a $\$5{,}000$ net amount at risk in addition to the account value. Find the expected number of deaths, voluntary lapses, depletion terminations, and policies remaining in force. Also calculate the probability of remaining in force for one policy.

Method. Apply the conditional probabilities in their stated order. The expected number in each group is the number reaching that stage multiplied by the applicable probability.

Expected deaths are $$ 1{,}000(0.02)=20. $$ The expected number surviving the year is $$ 1{,}000(1-0.02)=980. $$ Expected voluntary lapses among those survivors are $$ 980(0.08)=78.4. $$ The expected number who survive and avoid voluntary lapse is $$ 980-78.4=901.6, $$ which is also $980(0.92)$. Expected depletion terminations are $$ 901.6(0.05)=45.08. $$ Therefore the expected number in force at the start of the next year is $$ 901.6-45.08=856.52. $$ For an individual policy, the probability of remaining in force is $$ (1-0.02)(1-0.08)(1-0.05) =0.98(0.92)(0.95)=0.85652. $$ Multiplying by 1,000 gives $856.52$, consistent with the staged calculation. Expected death claims for the year are $20(\$5{,}000)=\$100{,}000$.

Check and common mistake avoided. The 8% lapse rate applies to the 980 survivors, not to all 1,000 start-of-year policies. Depletion applies only to the 901.6 expected survivors who also avoid voluntary lapse. Applying both rates directly to the original 1,000 would use the wrong exposure base.

Depletion Across More Than One Year

A policy that has a positive account at one year-end may still deplete later. Account projection and decrement projection therefore have to be linked year by year. The following example assumes no premiums, no death, and no voluntary lapse so that the account rule alone determines when coverage ends.

Worked Example: Finding the Year of Account Depletion

A hypothetical policy begins with $\$120$, receives no further premiums, earns a 1% annual credit, and incurs a fixed $\$50$ charge at each year-end. Coverage is maintained during a year if the account is positive at the start of that year. A nonpositive calculated balance terminates coverage at that year-end. Find the balances through termination.

Method. Apply $B_t=A_{t-1}(1.01)-50$ for each year in which the policy is in force at the start.

After year 1, $$ B_1=120(1.01)-50=121.20-50=\$71.20. $$ The account is positive, so the policy enters year 2. After year 2, $$ B_2=71.20(1.01)-50=71.912-50=\$21.912. $$ The account is still positive, so the policy enters year 3. After year 3, $$ B_3=21.912(1.01)-50=22.13112-50=-\$27.86888. $$ The calculated balance is nonpositive, so coverage ends at the end of year 3 under the stated rule. The negative calculated amount is a signal to apply the termination rule; it is not a negative account value carried into a fourth year.

Check and common mistake avoided. The year-2 balance is $\$21.912$, not a negative amount: the year-2 credit is $1\%$ of $\$71.20$, or $\$0.712$, before subtracting the $\$50$ charge. Applying the charge twice in year 2, or omitting that year’s credit, gives an incorrect depletion date.

Prospective Net Cash Flow and a Simple Capital Proxy

For a one-year prospective calculation, state which cash flows are included, when they occur, and to which policies they apply. In the convention used in the next example, the policy is in force at the start of the year; a death benefit equal to the net amount at risk is paid at year-end on death; a fixed expense and a fixed charge are both included at year-end for each start-of-year exposure. The charge is deducted immediately before any year-end death benefit is settled. These assumptions imply a one-year net outgo per policy in expectation of $q$ times the net amount at risk, plus the expense, minus the charge.

Definition: Under the stated one-year convention, with effective annual interest rate $i$, death probability $q$, net amount at risk $N$, year-end expense $E$, and year-end charge $C$, the prospective reserve for this one-year net cash flow is $$ V=\frac{qN+E-C}{1+i}. $$ This formula applies only to the cash flows and timing just specified. If charges are collected only from survivors, if a death claim includes a different benefit, or if there are later premiums, claims, or expenses, their expected present values must be included with their actual timing.

The numerator is the expected net outgo at year-end: death claims contribute $qN$, expenses contribute $E$, and charges reduce outgo by $C$. Because each amount is paid or received at year-end, discount the net amount once by $1+i$. This is a short one-year prospective valuation, not a complete multi-year reserve formula.

Worked Example: Calculating a One-Year Prospective Reserve

A hypothetical contract has a $\$3{,}600$ net amount at risk, a 2.5% probability of death during the year, a year-end expense of $\$18$, and a year-end charge of $\$60$. The effective annual interest rate is 4%. Under the convention above, calculate the one-year prospective reserve.

Method. First calculate expected death cost, then net the expense and charge, and finally discount the year-end net outgo.

Expected death cost is $$ 0.025(\$3{,}600)=\$90. $$ The expected net year-end outgo is $$ 90+18-60=\$48. $$ Discounting one year at 4% gives $$ V=\frac{48}{1.04}=\$46.153846\ldots. $$ To the nearest cent, the reserve is $\$46.15$.

Check and common mistake avoided. The charge reduces the expected outgo, but the net amount $\$48$ is still discounted at 4%. The calculation is not $\$37.50$; $\$48/1.04$ is approximately $\$46.15$.

Lapse and depletion also affect future claims and expenses by changing the number of policies exposed in later years. A simple scenario-based capital proxy can illustrate this effect, but it should not be confused with a regulatory capital requirement. Such a proxy compares specified stressed net outgo with a stated amount of available resources; it does not capture all required risk measures, diversification, or regulatory rules.

Worked Example: A Stressed One-Year Capital Proxy

A hypothetical block has 100 policies in force at the start of the year. Under a specified stress, the death probability is 3%, the net amount at risk is $\$4{,}000$, the year-end expense is $\$20$ per start-of-year policy, and the year-end charge is $\$35$ per start-of-year policy. Assume the charge is collected for each annual exposure as in the reserve convention above. The available resources for this one-year proxy are $\$7{,}500$. Separately, among policies surviving death, 15% voluntarily lapse; among those survivors who do not lapse, 10% deplete. Calculate the stressed net outgo, the proxy shortfall, and the expected number entering the next year.

Method. Calculate the one-year net outgo per start-of-year policy under the stated charge convention. Compare the block total with resources, then apply the sequential decrement probabilities to the starting count.

Expected death cost per start-of-year policy is $$ 0.03(\$4{,}000)=\$120. $$ Net year-end outgo per policy is $$ 120+20-35=\$105. $$ For 100 policies, stressed net outgo is $$ 100(\$105)=\$10{,}500. $$ The proxy shortfall relative to available resources is $$ 10{,}500-7{,}500=\$3{,}000. $$ For next-year exposure, the expected number surviving death is $$ 100(0.97)=97. $$ The expected number avoiding voluntary lapse is $$ 97(0.85)=82.45. $$ After allowing for depletion, the expected number entering the next year is $$ 82.45(0.90)=74.205. $$

Check and common mistake avoided. The shortfall is the excess of the specified stressed net outgo over the stated resources, not the whole $\$10{,}500$. The next-year count applies lapse to survivors and depletion only to survivors who avoid lapse. The $\$3{,}000$ is only a proxy under the assumptions in this example, not a statement of a regulatory capital requirement.

Common Modeling Traps

Dropping depletion from persistence. The expression $(1-q_t)(1-l_t)$ counts policies that survive and avoid voluntary lapse. It does not account for account depletion unless depletion is impossible.
Using the wrong conditional base. If lapse is conditional on surviving, apply $l_t$ to survivors, not to all start-of-year policies. Apply $d_t$ to the population specified by its conditional definition.
Ending current-year exposure at a year-end lapse. A policy lapsing at year-end was still exposed to the year’s death risk under this annual timing convention.
Carrying a negative account forward. A nonpositive calculated balance triggers the stated termination rule. Do not use that negative amount as the next year’s opening account unless the contract explicitly permits it.
Treating account value as reserve. Account value is determined by the contract’s account mechanics. A reserve is a valuation of specified future net cash flows under stated assumptions; the two quantities need not be equal.
Using a reserve formula without its timing assumptions. A charge collected only from survivors, for example, has a different expected value from a charge counted for every start-of-year exposure.

Key takeaway: Project the account, define the event order, and identify each conditional probability before calculating persistence or value. Lapse and depletion remove future exposure only from the time they occur; they do not erase the policy’s earlier exposure during the year.

Check Your Understanding

For each problem, use the timing and conditional-probability conventions stated in the question.

A policy starts with an account of $\$800$, receives a $\$200$ premium subject to a 5% load, earns 4%, and has $\$70$ of year-end charges. Calculate the year-end account and state whether depletion occurs. (Answer: $\$959.60$; no depletion.)
A group of 1,000 policies has a 3% death probability, a 10% voluntary lapse probability conditional on survival, and a 20% depletion probability conditional on surviving and avoiding lapse. Calculate the probability of entering the next year in force and the expected number doing so. (Answer: $0.6984$; $698.4$ policies.)
A one-year contract has a $\$6{,}000$ net amount at risk, death probability 2%, year-end expense $\$25$, and year-end charge $\$55$, under the one-year convention in this tutorial. At $i=5\%$, calculate the prospective reserve to the nearest cent. (Answer: $\$85.71$.)
A policy starts with $\$250$, has no premiums, earns 2% annually, and incurs a $\$80$ charge at each year-end. It remains in force during a year only if its opening account is positive; a nonpositive calculated balance terminates coverage at that year-end. Find the calculated balances for years 1 through 4 and the depletion year. (Answer: $\$175.00,\ \$98.50,\ \$20.47,\ -\$59.1206$; depletion occurs at the end of year 4.)
A block of 50 policies has a stressed death probability of 4%, a $\$3{,}000$ net amount at risk, a $\$40$ year-end expense and a $\$25$ year-end charge per start-of-year policy. Available resources are $\$5{,}000$. Calculate the one-year proxy shortfall. If 5% of survivors lapse and 10% of the remaining survivors deplete, calculate the expected number entering the next year. (Answer: $\$1{,}750$ shortfall; $41.04$ policies.)

What You'll Learn